Let $a = \frac{1 + \sqrt{2009}}{2}$ . Find the value of $(a^3 - 503a - 500)^5$ .

$a$ is a root of a quadratic equation with roots $$\frac{1 \pm \sqrt{2009}}{2}$$ That is, $a$ satisfies the following equation: $$x^2 - x - 502 = 0 \tag 1$$ Using this, we observe $$\begin{align}(a^3 - 503a - 500)^5 &= (a(\color{red}{a^2})-503a-500)^5 \\&\overset 1= (a(\color{red}{a+502})-503a-500)^5 \\&= (\color{blue}{a^2-a}-500)^5 \\&\overset 1= (\color{blue}{502} - 500)^5 \\&= 32 \end{align}$$

Let $a = \frac{1 + \sqrt{2009}}{2}$ . Find the value of $(a^3 - 503a - 500)^5$ .

I see no reason for elegance. Since one of the factors in the numerator is $1$, computing $a^3$ is is straightforward.

$$a^3 = \left(\frac{1}{8}\right) \times \left[ 1 + 3\sqrt{2009} + 3(2009) + 2009\sqrt{2009} \right]$$ $$=~ \left(\frac{1}{8}\right) \times \left[6028 + 2012\sqrt{2009}\right] ~=~ \frac{1507 + 503\sqrt{2009}}{2}. $$

Therefore,

$$(a^3 - 503a - 500)$$

$$=~ \frac{1507 + 503\sqrt{2009}}{2} ~-~ \frac{503 + 503\sqrt{2009}}{2} - \frac{1000}{2} ~=~ \frac{4}{2} \implies $$

$$(a^3 - 503a - 500)^5 = 2^5 = 32.$$